Fluctuations of the Boundary-Driven Symmetric Zero-Range Process from the NESS

We study the non-equilibrium stationary fluctuations of a symmetric zero-range process on the discrete interval $\{1, \ldots, N-1\}$ coupled to reservoirs at sites $1$ and $N-1$, which inject and remove particles at rates proportional to $N^{-\theta}$ for any value of $\theta\in\mathbb{R}$. We prove that, if the jump rate is bounded and under diffusive scaling, the fluctuations converge to the solution of a generalised Ornstein-Uhlenbeck equation with characteristic operators that depend on the stationary density profile. The limiting equation is supplemented with boundary conditions of Dirichlet, Robin, or Neumann type, depending on the strength of the reservoirs. We also introduce two notions of solutions to the corresponding martingale problems, which differ according to the choice of test functions.